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In this thesis we develop reduced dimensional (depth-averaged) models of the single and multi-phase flows of viscous fluids through Hele-Shaw channels with variable cross sections. We study the significance of including different physical properties in these models, specifically lateral viscous stress effects. Single-phase flow in uniform channels with an extreme width to depth aspect ratio (Hele-Shaw channels) that are modelled with an expression of Darcy's law produces an approximation of the depth-average of the solution to the Stokes equations. This approximation has been sufficiently accurate in much of the published work over the past century. However, this Darcy model is unable to correctly model problems that are dependent on lateral viscous stress effects, such as flow near any lateral walls or channel obstructions. Alternative depth-averaged `Stokes-Brinkman' models introduce lateral stress effects with ad hoc methods, rather than using a systematic approach, and currently the Stokes-Brinkman models in this thesis cannot be derived for non-uniform Hele-Shaw channels. We present the formal derivation of a novel `parabola-weighted' model that includes lateral stress effects in both uniform and non-uniform Hele-Shaw channels. In single-phase problems our parabola-weighted model is shown to approximate the three-dimensional Stokes solution as accurately as the Darcy model away from walls and obstacles, and better than or equal to the Stokes-Brinkman models outlined in this thesis in lateral stress dominated regions. In the multi-phase problem of modelling steadily propagating bubbles in uniform Hele- Shaw channels, the speed and shape of experimentally observed bubbles are correctly pre- dicted with the Darcy model, with exceptions in the small bubble limit and at low surface tensions. We explore the effect of lateral fluid stresses on these exceptions by modelling them with our parabola-weighted model. We show that lateral stress modelling has no significant effect on the bubbles predicted in domains where the Darcy model is known to accurately describe bubbles, as expected. However, these stresses are shown to be essential when modelling the small bubble limit, and we show that they slow bubbles with small surface tensions. Finally, the effect of lateral stress modelling in non-uniform Hele-Shaw channels oc- cluded by a rail is explored. In this domain, the range and interconnectivity of steady bubble solutions are known to be complex and there are limited experimental results. By comparing solutions to our parabola-weighted model to Darcy model solutions, we show that the additional lateral fluid stress effects predict new symmetry-breaking bifurcations and significant quantitative differences in the predicted bubble speed and critical point location for the experimentally unexplored solution branches.

Dried blood spot (DBS) is a blood collecting and storage method, which has been widely applied for newborn screening, therapeutic drug monitoring (TDM), and others. The concept of DBS involves the use a porous filter paper to absorb blood drops and preserve them as dried blood samples for further analysis. However, there are some disadvantages of the DBS process, such as, unequal distribution of the analyte of interest, haematocrit effects and difficulties in accurate analyte extraction, all of which have limited their applications for highly sensitive analysis. In addressing these issues, the spreading/imbibition of blood during the DBS sampling process has been investigated from both experimental and theoretical points of view in this work. DBS is described as a spreading process of a non-Newtonian drop (blood) over porous substrates (filter paper). Porcine blood with different haematocrit levels of spreading/penetrating over different porous substrates, including, commercial DBS filter papers and nitrocellulose membranes, have been used for the investigation via our spreading experiment. The time evolution of spreading parameters of DBS sampling process, such as, contact angles, droplet base radius and wetted region radius, were recorded and analysed to describe the spreading/imbibition behaviour of DBS sampling. The experimental results have shown that the spreading parameters of DBS sampling on each porous substrate fall into a universal curve under the dimensionless scale. According to our observations, the spreading behaviour of blood can be described as a complete wetting on the commercial DBS cards (Whatman 903) and partial wetting on the nitrocellulose membrane. Simulating models have been developed to describe the spreading/imbibition behaviours of DBS sampling process over porous substrate in complete wetting case. In the case of complete wetting, a system of two differential equations is derived, which describes the time evolution of radii of both the drop base and the wetted region inside the porous medium. The results show a good agreement while validating the spreading parameter, known as, droplet base radius, wetting region radius and contact angle with experimental data. These models provide simple and reliable methods to describe the blood spreading behaviour during DBS sampling process. In this project, the investigation of spreading behaviour of DBS analysis has been developed and presented. Reliable theoretical prediction of DBS sampling result can improve the defect of current detecting method and enhance the accuracy of the DBS analysis. In the consideration of the practical application of DBS analysis, the theoretical investigation of spreading/imbibition behaviour of DBS sampling should allow us to better control and predict the performance and outcome of pharmaceutical and analytical studies while using DBS method.

A vuggy porous medium is one with many small cavities called vugs, which are interconnected in complex ways forming channels that can support high flow rates. Flow in such a medium can be modeled by combining Darcy flow in the rock matrix with Stokes flow in the vugs. We develop a finite element for the numerical solution of this problem in three dimensions, which converges at the optimal rate. We design a multigrid method to solve a saddle point linear system that comes from this discretization. The intertwining of the Darcy and Stokes subdomains in a natural vuggy medium makes the resulting matrix highly oscillating, or ill-conditioned. The velocity field we are trying to compute is also very irregular and its tangential component might be discontinuous at the Darcy-Stokes interface. This imposes a difficulty in defining intergrid transfer v operators. Our definition is based on mass conservation and the analysis of the orders of magnitude of the solution. A new smoother is developed that works well for this ill-conditioned problem. We prove that coarse grid equations at all levels are well posed saddle point systems. Our algorithm has a measured convergence factor independent of the size of the system. We then use our solver to study transport and flow properties of vuggy media by simulations. We analyze the results of our transport simulations and compare them to experimental results. We study the influence of vug geometry on the macroscopic flow properties of a three dimensional vuggy porous medium.

Convective heat transfer during hot pressing in wood-based composite panel manufacturing is widely accepted as the most important means of heat transport for resin curing. The rate of convective heat transfer to the panel core is controlled by its permeability. Permeability in the plane of the panel also controls the flow of vapor to the panel edges, thereby influencing the potential for panel "blowing". This research considers how flake thickness, flake alignment and changing mat density during hot-pressing influences OSB mat permeability, through its thickness and in the plane of the panel. Some previous research exists but it fails to address the affects of horizontal and vertical density gradients as well as flake alignment. An apparatus was designed to allow cold pressing of aligned flakes to desired densities while enabling permeability measurements through the mat thickness. An additional apparatus was designed to allow the measuring of permeability in the plane of the mat. These designs permitted permeability measurements in mats that had no vertical density gradient, allowing for the direct study of permeability versus density (compaction ratio). Superficial permeability was determined using Darcy's law and for each sample, multiple readings were made at five different pressure differentials. Permeability through the mat thickness was highly dependent on compaction ratio and to a lesser extent flake thickness. As the compaction ratio is increased, the initial reduction in permeability is severe, once higher compaction ratios are achieved the reduction in permeability is less pronounced. Permeability decreased with decreasing flake thickness. Permeability in the plane of the mat decreases with increasing compaction ratio but in a less severe manner than through the mat thickness. In this case, the permeability-compaction ratio relationship appears linear in nature. Again, permeability decreases with decreasing flake thickness.

Study of Permeability of Pavement Base Materials